Abstract
In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincaré series. This relates to Rademacher’s “expansion of zero” principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.
References
Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass forms and mock modular forms: theory and applications, AMS Colloquium Series (to appear)
Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)
Fay, J.D.: Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 293/294, 143–203 (1977)
Knopp, M.I.: Construction of automorphic forms on H-groups and supplementary Fourier series. Trans. Am. Math. Soc. 103, 168–188 (1962)
Lawrence, R., Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. Math. 3, 93–108 (1999)
Lehner, J.: Partial fraction decompositions and expansions of zero. Trans. Am. Math. Soc. 87, 130–143 (1958)
Lehner, J.: Discontinuous Groups and Automorphic Functions. Mathematical Surveys, vol. VIII. American Mathematical Society, Providence, RI (1964)
Niebur, D.: A class of nonanalytic automorphic functions. Nagoya Math. J. 52, 133–145 (1973)
Ono, K.: Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory. Current Developments in Mathematics, vol. 2008, pp. 347–454. International Press, Somerville, MA (2009)
Rademacher, H.: A convergent series for the partition function p(n). Proc. Natl. Acad. Sci. U. S. A. 23(2), 78–84 (1937)
Rademacher, H.: Topics in Analytic Number Theory. Springer, New York/Heidelberg (1973). Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathemathischen Wissenschaften, Band 169
Rhoades, R.C.: A unified approach to partial and mock theta functions. Math. Res. Lett. (to appear)
Zagier, D.: Ramanujan’s mock theta functions an their applications (d’après Zwegers and Bringmann-Ono). Ásterique 326: Exp. No. 986, vii–viii, 143–164 (2010). 2009. Séminaire Bourbaki. Vol. 2007/2008
Zwegers, S.P.: Mock theta functions, Ph.D. thesis, Universiteit Utrecht (2002)
Acknowledgements
The research of the second author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant agreement n. 335220—AQSER.
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Andersen, N., Bringmann, K., Rolen, L. (2017). Images of Maass-Poincaré Series in the Lower Half-Plane. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_2
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DOI: https://doi.org/10.1007/978-3-319-69712-3_2
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